Method for equitable placement of a limited number of sensors for wide area surveillance

ABSTRACT

A limited number of sensors are placed at selected locations in order to achieve equitable coverage levels to all locations that need to be monitored. The coverage level provided to any specific location depends on all sensors that monitor the location and on the properties of the sensors, including probability of object detection and probability of false alarm. These probabilities may depend on the monitoring and monitored locations. An equitable coverage to all locations is obtained by finding the lexicographically largest vector of coverage levels, where these coverage levels are sorted in a non-decreasing order. The method generates a lexicographic maximin optimization model whose solution provides equitable coverage levels. In order to facilitate computations, a nonlinear integer optimization model is generated whose solution provides the same coverage levels as the lexicographic maximin optimization model. Solution of the nonlinear integer optimization model is obtained through the adaptation of known optimization methods.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Patent Application No. 60/901,909, filed Feb. 16, 2007, which is hereby incorporated herein by reference in its entirety.

FIELD OF INVENTION

The present invention relates to optimal placement of a limited number of sensors at selected locations in order to provide adequate protection to all locations.

BACKGROUND OF THE INVENTION

The use of sensors to provide effective surveillance of wide areas is becoming increasingly common. Consider a specified area where harmful objects may be placed, such as explosives, biological agents, or chemical substances. A fixed number of sensors are installed throughout the area, where each of these sensors provides observations on one or more locations within the area. The observations of these sensors are combined through a data fusion process in order to assess whether an object is actually present at one or more of the observed locations or not. Since the number of sensors that can be placed is limited, it is critically important to determine optimal locations for these sensors. In some applications, many sensors may be installed, but only a limited number of these sensors can be activated simultaneously.

Sensors are also used for intrusion detection. Defense against intrusion may be necessary to protect large areas like a border between countries, oil and gas pipelines, strategic facilities like nuclear reactors, industrial complexes, military bases, etc. Again, placing the sensors optimally is vitally important in order to achieve appropriate protection against intruders who might approach the protected area from different directions.

A related topic focuses on the optimal location of emergency facilities, such as emergency rooms, fire departments, and police stations. It is convenient to represent an area by a network, where each node represents a neighborhood, e.g., a square of dimension 100×100 meters. A link interconnecting a pair of nodes represents possible movement from one node to the other and the link metric represents the distance (or travel time) between the end-nodes. A typical problem is to place a limited number of emergency facilities at a subset of these nodes so that the distance (or travel time) from any node to the closest facility is minimized. This is a well-known problem in the literature, referred to as the network minimax location problem or the vertex center problem. A related problem, known in the literature as the set covering problem, minimizes the cost of installing facilities at a subset of the nodes so that each of the nodes is within a specified distance (or travel time) from the closest facility. L. V. Green and P. J. Kolesar, “Improving Emergency Responsiveness with Management Science”, Management Science, 50, 1001-1014, 2004 present the state-of-the-art of emergency responsiveness models.

The optimal locations of emergency facilities under the network minimax location problem are not unique as there may be numerous solutions that provide the best possible service to the worst-off location. Hence, it would be attractive to find which solution from among all minimax solutions should be selected. W. Ogryczak, “On the Lexicographic Minimax Approach to Location Problems”, European Journal of Operational Research, 100, 566-585, 1997 presents an algorithm to find a lexicographic minimax solution to the location problem. As in the minimax network location problem, any specific location is served by a single facility, specifically, by the closest facility to that location. The lexicographic minimax solution is the best minimax solution in the sense that ordering the service provided to the locations (in terms of distance or travel time from closest facility) from the worst to the best, the resulting ordered vector is the lexicographically smallest possible ordered vector. Such a solution is referred to as an equitable solution.

K. Chakrabarty, S. S. Iyengar, H. Qi, and E. Cho, “Grid Coverage for Surveillance and Target Location in Distributed Sensor Networks,” IEEE Transactions on Computers, 51, 1448-1453, 2002 formulate a sensor location problem as a set covering problem which minimizes the cost of installing sensors at a subset of the nodes so that each of the nodes is within a specified distance (or travel time) from a specified number of sensors.

This invention focuses on placing a limited number of sensors in order to achieve an equitable coverage of all locations, using a lexicographic maximin objective. The coverage level provided to any specific location may depend on the locations of multiple sensors that monitor the location and on the properties of the sensors. This is a significant extension of the paper above by W. Ogryczak, and the method used there cannot be extended to solve the problem addressed by this invention. H. Luss, “On Equitable Resource Allocation Problems: A Lexicographic Minimax Approach”, Operations Research, 47, 361-378, 1999 provides an exposition of various equitable resource allocation models and solution methods; however, none of these can be applied to this invention.

SUMMARY OF INVENTION

The present invention focuses on placing a limited number of sensors at selected locations in order to achieve equitable coverage levels to all locations that need to be monitored. The area under surveillance is represented as a network where the nodes represent locations and the interconnecting links indicate surveillance relations. Consider a sensor at node j. In addition to monitoring node j, a link from node j to node i means that a sensor at node j can also monitor node i. The coverage level provided to any specific location depends on all sensors that monitor that location and on the properties of the sensors. The properties of a sensor include the probability of detecting a target at a specified location when a target is present at that location and the probability of erroneously detecting a target at same location when a target is not present there. These probabilities may be different for each (i, j) node-pair.

Suppose the locations of the sensors are specified. Given these locations, the coverage level offered to each location is computed. Consider the vector of the coverage levels offered to each of the locations where the elements of this vector (i.e., the coverage levels) are sorted in a non-decreasing order. Equitable coverage levels to all locations are specified as the lexicographically largest such ordered vector of coverage levels. The invention determines optimal locations of a limited number of sensors so that equitable coverage levels to all locations are achieved. The invention generates the equitable sensor location model as a lexicographic maximin optimization model whose solution provides equitable coverage levels to all locations. Current state-of-the-art of optimization solvers cannot directly solve said lexicographic maximin optimization model. The invention generates a nonlinear integer optimization model whose solution would also provide equitable, or near-equitable, coverage levels to all locations. Solution of said nonlinear integer optimization model can be obtained through the adaptation of known optimization methods, such as dynamic programming and various meta-heuristics, including simulated annealing and tabu search. The sensor location model can be part of a system used in a static (one time) situation or a dynamic (multi-period) situation. In a dynamic situation, sensor locations are periodically changed to prevent learning of the locations by an adversary.

The present invention will be more clearly understood when the following description is read in conjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a network representation of the sensor location model.

FIG. 2 illustrates a bipartite network representation of the sensor location model.

FIG. 3 is a flow chart of a method for determining sensor locations that provide equitable coverage levels to all locations.

DETAILED DESCRIPTION

Referring now to the figures and FIG. 1 in particular, there is shown an example of a network 100 representing an area under surveillance. The area is represented by six nodes 101-106. Three sensors are located as represented by the nodes in black 101, 104 and 106. The optimal location of the sensors will be determined in accordance with the teachings of the present invention. The directed links 107-115 represent surveillance relations. For example, 107 represents surveillance relations from node 101 to node 102 (shown by the solid directed link) and from node 102 to node 101 (shown by the dashed directed link). The solid directed link indicates that indeed node 101 has a sensor that monitors node 102. The dashed directed link indicates that node 102 could have monitored node 101 if a sensor had been placed at node 102. Note that, for example, the sensor at node 101 monitors nodes 101, 102, 105 and 106 and that node 101 is monitored by the sensors at nodes 101 (a sensor always monitors the node in which it is located) and 106.

The following notation is used:

N=Set of nodes that need to be monitored. Nodes in N are indexed by i. In FIG. 1, N={101, 102, 103, 104, 105, 106). In practical situations N may be large where the specific value depends on the area size and on the area represented by a node. For instance, if the area under surveillance is a square of 10 km×10 km and each node represents a square of 100 m×100 m, N=10,000. S=Set of nodes where sensors can be located. Nodes in S are indexed by j. Although in example 100 it is assumed that S=N, the sets N and S may not be the same. J(i)=Subset of nodes in S that can monitor node i. The set J(i) includes all nodes that have a link directed into node i plus, if iεS, node i itself. For example, in FIG. 1 J(101)={101, 102, 105, 106} and J(102)={101, 102, 103, 106}.

The present invention provides a method that determines optimal locations when the number of available sensors is limited. Although the sensors are assumed to be identical, the coverage level that a sensor placed at node j provides to node i depends on the sensor properties and on the specified nodes i and j. The sensor properties are typically specified through the following probabilities:

p_(ij)=Probability that a sensor at node j detects an object at node i, given that there is an object at node i. q_(ij)=Probability that a sensor at node j erroneously detects an object at node i, given there is not an object at i (false alarm). We assume that q_(ij)<p_(ij).

The coverage level provided to location i is determined based on all sensors located at nodes that are in set J(i). An optimal solution to the sensor location problem will be a solution that provides equitable coverage to all nodes in N. An equitable coverage solution will be defined later.

In FIG. 2 there is shown example of a different network 200 representation of the sensor location model wherein the model is shown as a bipartite network. Nodes 201-206 are the set of nodes S where sensors can be placed. These nodes correspond to nodes 101-106 in network 100. Sensors are located in network 200 at nodes 201, 204 and 206 (nodes in black), corresponding to sensor locations at nodes 101, 104 and 106 in network 100. Each of the nodes 201-206 is duplicated on the right side of network 200. Thus, node 207 is a duplicate of node 201, node 208 is a duplicate of node 202, etc. Nodes 207-212 represent the set of nodes N that needs to be monitored. Although in this example the sets N and S include the same nodes (same assumption as made in network 100), this need not be the case. If S and N are not the same, some nodes in S may not have duplicate nodes in N and some nodes in N may not have corresponding nodes in S. The links in network 200 indicate surveillance relations. Thus, for example, node 201 has links 213 a-d to nodes 207, 208, 211 and 212, respectively, and node 202 has links 216 a-d to nodes 207, 208, 209 and 212, respectively. These links have obvious one-to-one correspondence to links in network 100 with the addition of a link from a node in set S to its duplicate node in set N. Note that the solid links connect a node with a sensor to the relevant nodes in N and the dashed links connect a node without a sensor to the relevant nodes in N. The sets J(i) are readily derived from network 200, for example J(207)={201, 202, 205, 206}. Hence, in this example node 207 is monitored by two sensors in nodes 201 and 206}. Note that J(207) corresponds uniquely to J(101)={101, 102, 105, 106} in network 100 where node 101 is monitored by the two sensors at nodes 101 and 106.

FIG. 3 presents a flow chart of a method 300 for determining sensor locations that provide equitable coverage levels to all locations.

Input Preparation to the Method (Steps 301 and 302)

A network representation of the specified area is generated 301, as explained above with reference FIGS. 1 and 2. The number of nodes in the network depends on the area size and on the area represented by a single node. The required accuracy depends upon the specific applications. Characterization of properties of the sensors that would affect quality of surveillance is specified 302. These properties may include, but are not limited to, probabilities p_(ij) and q_(ij). Although all sensors are assumed to be of the same type, note that these probabilities depend on the sensor location and the monitored location. Different probabilities from different sensor locations may result from different distances of these locations to node i or from some obstacles between these locations and nodes i. The network representation (either the network 100 of FIG. 1 or the network 200 of FIG. 2) and the properties of the sensors are the primary inputs to the sensor location model.

Generation of Surveillance Performance Functions (Step 303)

The goal of the method is to determine optimal locations of a limited number of available sensors. Let

x_(j)=Decision variable. x_(j)=1 if a sensor is located at node j and x_(j)=0 otherwise. Let x be the vector of all decision variables x_(j), jεS. ƒ_(i)(x)=Surveillance performance function for node i, iεN. For a specific value of the vector x, resulting value of this function is also referred to as the coverage level provided to node i. Note that the only decision variables x_(j) that affect ƒ_(i)(x) are those for which jεJ(i).

Two examples are provided below for possible surveillance performance functions. The invention is not limited to these specific performance functions.

EXAMPLE 1

Suppose all probabilities q_(ij)=0 and 0<p_(ij)<1. Then, the surveillance performance function for node i, as a function of x, can be set to the probability that an object at node i will be detected by at least one sensor. This implies

$\begin{matrix} {{{f_{i}(x)} = {1 - {\prod\limits_{{x_{j} = 1},{j \in {J{(i)}}}}\left( {1 - p_{ij}} \right)}}},{i \in N},{{{where}\mspace{14mu} {\prod\limits_{\varphi}( \cdot )}} = 1.}} & (1) \end{matrix}$

The value of ƒ_(i)(x) for a specified x is referred to as the coverage level provided to node i; the larger the value of ƒ_(i)(x), the better is the coverage level provided to node i. Note that equation (1) can also be written as

${{f_{i}(x)} = {1 - {\prod\limits_{j \in {J{(i)}}}\left( {1 - p_{ij}} \right)^{x_{j}}}}},{i \in {N.}}$

EXAMPLE 2

Suppose all probabilities q_(ij) satisfy 0<q_(ij)<p_(ij). Then, the effectiveness of a sensor t location j for detecting an object at location i can be estimated by the ratio (1-p_(ij))/(1-q_(ij)); the smaller this ratio, the more effective the sensor. Obviously, if q_(ij) is about equal to p_(ij), placing a sensor at node j to monitor node i is useless as the collected information from that sensor would not provide any meaningful information. Note that

$\prod\limits_{{x_{j} = 1},{j \in {J{(i)}}}}\left( {1 - p_{ij}} \right)$

is the conditional probability that none of the sensors that monitor node i would detect an object at node i given that there is an object at node i, and

$\prod\limits_{{x_{j} = 1},{j \in {J{(i)}}}}\left( {1 - q_{ij}} \right)$

is the conditional probability that none of the sensors that monitor node i would erroneously detect an object at node i given that there is no object at node i. Similarly to equation (1), one minus the ratio of these conditional probabilities can be selected to form the following surveillance performance function.

$\begin{matrix} {{{f_{i}(x)} = {1 - {\prod\limits_{{x_{j} = 1},{j \in {J{(i)}}}}\frac{1 - p_{ij}}{1 - q_{ij}}}}},{i \in N},{{{where}\mspace{14mu} {\prod\limits_{\varphi}( \cdot )}} = 1.}} & (2) \end{matrix}$

Note that equation (2) can also be written as

${{f_{i}(x)} = {1 - {\prod\limits_{{j \in J} = {(i)}}\left( \frac{1 - p_{ij}}{1 - q_{ij}} \right)^{x_{j}}}}},{i \in {N.}}$

Various other surveillance functions can be used. Consider a specific i, and let x¹ and x² be two vectors where x_(j) ¹≦x_(j) ² for all jεJ(i) and x_(j) ¹<x_(j) ² for at least one jεJ(i). The surveillance performance functions should satisfy the following properties:

Property (i)

The function ƒ_(i)(x) is increasing with variables jεJ(i), i.e., ƒ_(i)(x¹)<ƒ_(i)(x²).

Property (ii)

Suppose some variable jεJ(i) that is 0 in x¹ and in x² is set to 1 in both x¹ and x², resulting vectors x¹⁺ and x²⁺, respectively. Then, ƒ_(i)(x¹⁺)−ƒ_(i)(x¹)≧ƒ_(i)(x²⁺) ƒ_(i)(x²); i.e., ƒ_(i)(x) is concave on the integer values of x_(j) for jξJ(i).

Note that properties (i) and (ii) hold for equations (1) and (2).

Generation of Equitable Sensor Location Model—ESLM (Step 304)

The model is formulated with surveillance performance functions ƒ_(i)(x) for iεN.

Let

ƒ^((n))(x)=Vector of all ƒ_(i)(x)'s, sorted in non-deccreasing order, that is,

ƒ^((n))(x)=[ƒ_(i) ₁ (x), ƒ_(i) ₂ (x), . . . , ƒ_(i) _(|N|) (x)],  (3.a)

where

ƒ_(i) ₁ (x)≦ƒ_(i) ₂ (x)≦ . . . ≦ƒ_(i) _(|N|) (x).  (3.b)

P=Number of sensors available, P<|S|. These sensors are placed at a subset of the nodes in the set of nodes S, at most one sensor per node. The case P≧|S| need not be considered as it results in a trivial problem where a sensor is placed at each of the nodes in the set S.

An equitable solution is a solution that provides the lexicographic largest vector ƒ^((n))(x). The Equitable Sensor Location Model, referred to as ESLM, is formulated as a lexicographic maximin optimization model.

ESLM

$\begin{matrix} {{V^{f} = {{lex}\; {\max\limits_{x}\left\lbrack {f^{(n)}(x)} \right\rbrack}}}{{So}\mspace{14mu} {that}}} & \left( {4.a} \right) \\ {{{f^{(n)}(x)} = \left\lbrack {{f_{i_{1}}(x)},{f_{i_{2}}(x)},\ldots \mspace{11mu},{f_{i_{N}}(x)}} \right\rbrack},} & \left( {4.b} \right) \\ {{{f_{i_{1}}(x)} \leq {f_{i_{2}}(x)} \leq \ldots \leq {f_{i_{N}}(x)}},} & \left( {4.c} \right) \\ {{{\sum\limits_{j \in S}x_{j}} = P},} & \left( {4.d} \right) \\ {{x_{j} = 0},1,{j \in {S.}}} & \left( {4.e} \right) \end{matrix}$

Objective function (4.a) finds the lexicographic largest vector V^(ƒ), where by statements (4.b) and (4.c) this vector comprises all surveillance performance functions ƒ_(i)(x) sorted in a non-decreasing order. Constraints (4.d) and (4.e) limit the number of placed sensors to P where at each node of the set S at most one sensor is placed. As discussed above, examples of surveillance performance functions are given in equations (1) and (2). ESLM is independent of the specific form used for the surveillance performance functions as long as these functions are increasing (property (i) in step 303)

Generation of Executable Equitable Sensor Location Model (Step 305)

Although ESLM provides a complete and accurate formulation for computing equitable solutions, this formulation cannot be solved directly by known optimization methods.

Since, as discussed above, it is assumed that each of the surveillance performance functions ƒ_(i)(x) for iεN is an increasing function and concave on the integer values of x_(j) for jε J(i) (as specified by properties (i) and (ii) in step 303), an equitable solution (a lexicographic maximin solution) will be obtained by solving a related nonlinear integer optimization model. Note that the surveillance performance functions specified in equations (1) and (2) are given for illustrative purposes only; all that is required is that the functions satisfy properties (i) and (ii) specified in step 303. Let K be an arbitrarily large parameter. The solution of the following nonlinear integer optimization model would provide an equitable solution to the Equitable Location Sensor Model as formulated by ESLM. The new model is referred to as the Equitable Sensor Location Model—Executable (ESLM-EX).

ESLM-EX

$\begin{matrix} {{V^{K} = {\min\limits_{x}\left\{ {\sum\limits_{i \in N}\frac{1}{\left\lbrack {ɛ + {f_{i}(x)}} \right\rbrack^{K}}} \right\}}}{{so}\mspace{14mu} {that}}} & \left( {5.a} \right) \\ {{{\sum\limits_{j \in S}x_{j}} = P},} & \left( {5.b} \right) \\ {{x_{j} = 0},1,{j \in S},} & \left( {5.c} \right) \end{matrix}$

where ε is an arbitrarily small parameter introduced for computational purposes to avoid infinite terms in the objective function (5.a). When K is very large, ESLP-EX will provide an equitable solution, or, equivalently, a lexicographic maximin solution. Suppose ƒ₁(x)<ƒ₂(x). Then, property (i) in step 303 implies that for large K the term for i=1 in objective function (5.a) is significantly larger than the term for i=2 in objective function (5.a). This argument applies to every pair of nodes in N. Property (ii) in step 303 implies that the improvement in the i-th term in the objective function (5.a) is larger when x¹ is increased to x¹⁺ than the improvement realized when x² is increased to x²⁺. Thus, for a sufficiently large value K, an optimal solution of ESLM-EX would be the lexicographically largest feasible vector of the performance function values sorted in non-decreasing order. Note that even a small value of K (e.g., K≧4) the solution of ESLM-EX is expected to provide a near-equitable solution. An appropriate value of K can be determined through experimentation.

Computation of Equitable Solution (Step 306)

The present invention generates model ESLM-EX, whose solution provides an equitable solution to the Equitable Sensor Location Model, where the solution can be computed by various existing state-of-the-art optimization methods. These include, but are not limited to, dynamic programming and meta-heuristics such as simulated annealing and tabu search. The book by T. Ibaraki and N. Katoh, “Resource Allocation Problems: Algorithmic Approaches”, The MIT Press, Cambridge, Mass., 1988 provides in Section 3.2 a dynamic programming algorithm that solves ESLM-EX. C. R. Reeves (editor), “Modern Heuristic Techniques for Combinatorial Problems”, Halsted Press an imprint of John Wiley, New York, 1993, presents in his book tutorials on various meta-heuristics, including on simulated annealing and tabu search.

ESLM-EX may be used in a static (a single period) or a dynamic (multi-period) environment. Consider a dynamic environment where, for example, data is collected from all sensors every 15 minutes and the data analysis repeatedly suggests that no objects are present at any of the locations. Still, after some time, e.g., after a day, it is desirable to change some of the sensor locations so that an adversary would not be able to learn where sensors are located. This can be done, for example, by changing the set S of possible sensor locations and resolving ESLM-EX. The changes in the set S can be selected using some randomized selection scheme. In some applications, sensors are installed at every node in S, however, at every point in time, only P<|S| of these sensors are activated due to operational constraints. In such applications, the locations of activated sensors would be changed periodically by resolving ESLM-EX, wherein the set S is changed using some randomized selection scheme.

Finally, suppose the data analysis suggests that there is a concern that objects are present at a subset of the locations, say at subset of nodes N^(present). ESLM-EX can then be applied to a new network representation that includes explosion of the nodes in N^(present) so that each node in the new network would represent a much smaller area than in the original network. ESLM-EX would then find an equitable solution to place a limited number of a second type of sensors, e.g., mobile sensors, in the area represented by the new network in order to collect more accurate observations of the area under suspicion.

The algorithms and modeling described above are capable of being performed on an instruction execution system, apparatus, or device, such as a computing device. The algorithms themselves may be contained on a computer-readable medium that can be any means that can contain, store, communicate, propagate, or transport the program for use by or in connection with an instruction execution system, apparatus, or device, such as a computer.

While there has been described and illustrated a method for the optimal placement of a limited number of sensors at selected locations in order to achieve equitable coverage levels to all locations, it will be apparent to those skilled in the art that variations and modifications are possible without deviating from the broad teachings and scope of the present invention which shall be limited solely by the scope of the claims appended hereto. 

1. A method for determining optimal placement of a limited number of sensors in a specified area wherein optimal location of sensors provides equitable coverage levels to all locations within the area comprising the steps of: (a) generating a network representation of a specified area comprising nodes and directed links, where each node represents a sub-area that is to be monitored or a sub-area where a sensor may be placed or both and where each directed link represents surveillance relations among node-pairs, where a node-pair comprises a first node in a set of nodes where a sensor may be placed and an associated second node in a set of nodes that is to be monitored; (b) characterizing sensors in terms of their properties including probabilities of object detection and probabilities of false alarms, where the probabilities may be different for different node-pairs; (c) generating surveillance performance functions for the coverage level provided to each of the nodes that is to be monitored as a function of the location of sensors that monitor each of the nodes that is to be monitored; (d) generating an equitable sensor location model as a lexicographic maximin optimization model whose solution provides the lexicographically largest ordered vector whose elements are the coverage levels provided to the monitored nodes sorted in non-decreasing order, wherein the solution provides equitable coverage levels; and (e) generating the lexicographic maximin optimization model as a nonlinear integer optimization model whose solution provides equitable coverage levels to all nodes, where the solution is computed by existing optimization methods.
 2. A method for determining optimal placement of a limited number of sensors in a specified area wherein optimal location of sensors provides equitable coverage levels to all locations within the area comprising the steps of: (a) generating a network representation of a specified area comprising nodes and directed links, where each node represents a sub-area that is to be monitored or a sub-area where a sensor may be placed or both and where each directed link represents surveillance relations among node-pairs, where a node-pair comprises a first node in a set of nodes where a sensor may be placed and an associated second node in a set of nodes that is to be monitored; (b) characterizing sensors in terms of their properties including probabilities of object detection and probabilities of false alarms, where the probabilities may be different for different node-pairs; (c) generating surveillance performance functions for the coverage level provided to each of the nodes that is to be monitored as a function of the location of sensors that monitor each of the nodes that is to be monitored; (d) generating an equitable sensor location model ESLM as a lexicographic maximin optimization model whose solution provides the lexicographically largest ordered vector whose elements are the coverage levels provided to the monitored nodes sorted in non-decreasing order, wherein the solution provides equitable coverage levels; and (e) generating the lexicographic maximin optimization model as a nonlinear integer optimization model whose solution provides equitable coverage levels to all locations that need to be monitored within the area, where the solution is computed by solving ESLM-EX.
 3. The method as set forth in claim 2, where ESLM is a lexicographic maximin optimization model and the objective function specifies the lexicographically largest vector whose elements are coverage levels to the monitored sub-areas sorted in a non-decreasing order and constraints specify a limit on the number of sensors that are placed and decision variables represent sensor placement decisions.
 4. The method as set forth in claim 2, where the nodes comprise a node set N representing sub-areas that are to be monitored, node set S represents locations where sensors can be placed, and a link from a node in set S to an associated node in set N indicates that the node in set S can monitor the associated node in N in a node-pair.
 5. The method as set forth in claim 4, where surveillance performance functions compute the coverage level offered to any specified node in set N as a function of all sensors placed at nodes in set S that can monitor the specified node in set N.
 6. The method as set forth in claim 5, where the surveillance performance functions use as input properties of sensors, including probabilities of object detection and probabilities of false alarms, wherein the probabilities may be different for each node-pair in the sets of nodes N and S.
 7. The method as set forth in claim 2, wherein nonlinear integer optimization model ESLM-EX is generated from ESLM and the optimal solution of ESLM-EX provides equitable coverage levels to all nodes that are to be monitored.
 8. The method as set forth in claim 7, wherein the optimal solution of ESLM-EX is generated by applying known optimization methods.
 9. A method for determining optimal placement of a limited number of sensors in a set of nodes S that provides equitable coverage levels to nodes in a set of nodes N, (a) characterizing sensors in terms of their properties, including probabilities of object detection and probabilities of false alarms, wherein each node-pair has its associated probabilities; (b) generating surveillance performance functions for the coverage level provided to each of the nodes in set N as a function of the sensors placed at nodes in the set S that monitor an associated node in set N; (c) generating model ESLM as a lexicographic maximin optimization model whose solution provides the lexicographically largest ordered vector whose elements are the coverage levels provided to the nodes in set N sorted in non-decreasing order, and the solution provides equitable coverage levels to all nodes in set N; and (d) generating model ESLP-EX whose solution is computed by existing optimization methods providing equitable coverage levels to all nodes in set N.
 10. A method as set forth in claim 9, wherein the locations of at least one of the sensors is changed from one period to the next.
 11. The method as set forth in claim 10, wherein in each period ESLP-EX is solved with a modified set of possible sensor locations.
 12. A system for determining optimal placement of a limited number of sensors in a specified area wherein optimal location of sensors provides equitable coverage levels to all locations within the area comprising: (a) means for generating a network representation of a specified area comprising nodes and directed links, where each node represents a sub-area that is to be monitored or a sub-area where a sensor may be placed or both and where each directed link represents surveillance relations among node-pairs, where a node-pair comprises a first node in a set of nodes where a sensor may be placed and an associated second node in a set of nodes that is to be monitored; (b) a plurality of sensors characterized in terms of their properties, including probabilities of object detection and probabilities of false alarms, where the probabilities may be different for different node-pairs; (c) means for generating surveillance performance functions for the coverage level provided to each of the nodes that are to be monitored as a function of the location of sensors that monitor each of the nodes that is to be monitored; (d) means for generating an equitable sensor location model as a lexicographic maximin optimization model whose solution provides the lexicographically largest ordered vector whose elements are the coverage levels provided to the monitored nodes sorted in non-decreasing order, wherein the solution provides equitable coverage levels; (e) means for generating the lexicographic maximin optimization model as a nonlinear integer optimization model whose solution provides equitable coverage levels to all nodes, where the solution is computed by existing optimization methods; and (f) said plurality of sensors being placed in accordance with the solution of the lexicographic optimization model.
 13. A system for determining optimal placement of a limited number of sensors in a specified area wherein optimal location of sensors provides equitable coverage levels to all locations within the area comprising: (a) means for generating a network representation of a specified area comprising nodes and directed links, where each node represents a sub-area that is to be monitored or a sub-area where a sensor may be placed or both and where each directed link represents surveillance relations among node-pairs, where a node-pair comprises a first node in a set of nodes where a sensor may be placed and an associated second node in a set of nodes that is to be monitored; a plurality of sensors characterized in terms of their properties, including probabilities of object detection and probabilities of false alarms, where the probabilities may be different for different node-pairs; (b) means for generating surveillance performance functions for the coverage level provided to each of the nodes that are to be monitored as a function of the location of sensors that monitor each of the nodes that is to be monitored; (c) means for generating an equitable sensor location model ESLM as a lexicographic maximin optimization model whose solution provides the lexicographically largest ordered vector whose elements are the coverage levels provided to the monitored nodes sorted in non-decreasing order, wherein the solution provides equitable coverage levels; (d) means for generating the lexicographic maximin optimization model as a nonlinear integer optimization model ESLM-EX whose solution provides equitable coverage levels to all nodes, where the solution is computed by existing optimization methods; and (e) said plurality of sensors being placed in for providing equitable coverage levels to all locations that need to be monitored within the area wherein the optimal solution is the solution of ESLM-EX.
 14. The system as set forth in claim 13, where ESLM is a lexicographic maximin optimization model and the objective function specifies the lexicographically largest vector whose elements are coverage levels to the monitored sub-areas sorted in a non-decreasing order and constraints specify a limit on the number of sensors that are placed and decision variables represent sensor placement decisions.
 15. The system as set forth in claim 13, where the nodes comprise a node set N representing sub-areas that are to be monitored, node set S represents locations where sensors can be placed, and a link from a node in set S to an associated node in set N indicates that the node in set S can monitor the associated node in N in a node-pair.
 16. The system as set forth in claim 15, where surveillance performance functions compute the coverage level offered to any specified node in set N as a function of all sensors placed at nodes in set S that can monitor said specified node in set N.
 17. The system as set forth in claim 16, where the surveillance performance functions use as input properties of sensors, including probabilities of object detection and probabilities of false alarms, wherein the probabilities may be different for each node-pair in the sets of nodes N and S.
 18. The system as set forth in claim 13, wherein nonlinear integer optimization model ESLM-EX is generated from ESLM and the optimal solution of ESLM-EX provides equitable coverage levels to all nodes that are to be monitored.
 19. The system as set forth in claim 18, wherein the optimal solution of ESLM-EX is generated by applying known optimization methods.
 20. A system for determining optimal placement of a limited number of sensors in a set of nodes S that provides equitable coverage levels to nodes in a set of nodes N comprising: a. a plurality of sensors characterized in terms of their properties, including probabilities of object detection and probabilities of false alarms, wherein each node-pair has its associated probabilities; b. means for generating surveillance performance functions for the coverage level provided to each of the nodes in set N as a function of the sensors placed at nodes in the set S that monitor an associated node in set N; c. means for generating model ESLM as a lexicographic maximin optimization model whose solution provides the lexicographically largest ordered vector whose elements are the coverage levels provided to the nodes in set N sorted in non-decreasing order, and the solution provides equitable coverage levels to all nodes in set N; d. means for generating model ESLP-EX whose solution is computed by existing optimization methods providing equitable coverage levels to all nodes in set N; and e. said plurality of sensors being placed in accordance with the solution of the ESLP-EX model.
 21. The system as set forth in claim 20, wherein the locations of at least one of the sensors is changed from one period to the next.
 22. The system as set forth in claim 21, wherein in each period ESLP-EX is solved with a modified set of possible sensor locations. 